In order to work with these complex numbers without drawing vectors, we first need some kind of standard mathematical notation. There are two basic forms of complex number notation:
polar
and
rectangular
.

Polar form is where a complex number is denoted by the
length
(otherwise known as the
magnitude
,
absolute value
, or
modulus
) and the
angle
of its vector (usually denoted by an angle symbol that looks like this: ∠). To use the map analogy, polar notation for the vector from New York City to San Diego would be something like “2400 miles, southwest.” Here are two examples of vectors and their polar notations: (Figure
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)

Vectors with polar notations.

Standard orientation for vector angles in AC circuit calculations defines 0
o
as being to the right (horizontal), making 90
o
straight up, 180
o
to the left, and 270
o
straight down. Please note that vectors angled “down” can have angles represented in polar form as positive numbers in excess of 180, or negative numbers less than 180. For example, a vector angled ∠ 270
o
(straight down) can also be said to have an angle of -90
o
. (Figure
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) The above vector on the right (7.81 ∠ 230.19
o
) can also be denoted as 7.81 ∠ -129.81
o
.

The vector compass

Rectangular form, on the other hand, is where a complex number is denoted by its respective horizontal and vertical components. In essence, the angled vector is taken to be the hypotenuse of a right triangle, described by the lengths of the adjacent and opposite sides. Rather than describing a vector’s length and direction by denoting magnitude and angle, it is described in terms of “how far left/right” and “how far up/down.”

These two dimensional figures (horizontal and vertical) are symbolized by two numerical figures. In order to distinguish the horizontal and vertical dimensions from each other, the vertical is prefixed with a lower-case “i” (in pure mathematics) or “j” (in electronics). These lower-case letters do not represent a physical variable (such as instantaneous current, also symbolized by a lower-case letter “i”), but rather are mathematical
operators
used to distinguish the vector’s vertical component from its horizontal component. As a complete complex number, the horizontal and vertical quantities are written as a sum: (Figure
below
)

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Foreword by Yong Zhao

Other Titles in:

From the authors of the best-selling
Habits of Mind
…

A costly gap exists between what we claim to value as educational essentials and how we evaluate results. In
Dispositions
, Arthur L. Costa and Bena Kallick present a game-changing look at why and how to “mind the gap”.

Featuring techniques for integrating dispositional learning into the Common Core,
Dispositions
is Costa and Kallick’s most indispensable work on 21st Century learning.

“Education must include the development of the human elements in our children. Costa and Kallick present a reasonable, practical and positive alternative—the cultivation of dispositional thinking, the very core of being human.”
Foreword by Yong Zhao

“Education must include the development of the human elements in our children. Costa and Kallick present a reasonable, practical and positive alternative-- the cultivation of dispositional thinking, the very core of being human. “